I suggest you adopt the verbiage "locality" which is stricter term than causality. The corollaries to the postulate of relativity (aka the "special" theory of relativity) only stipulates that it is not possible to accelerate from under to above the speed of light in a finite time. Similarly...
In the penultimate expression by Vanhees71 the integral depends only on one time variable, so T becomes superfluous; simply integrate then exponentiate. Why keep T then?